Whakaoti i te x^2+7x+10=

$${ x }^{ 2 } +7x+10=$$

Tauwehe

$$\left(x+2\right)\left(x+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=1\times 10=10$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 10.

$$1,10 2,5$$

Tātaihia te tapeke mō ia takirua.

$$1+10=11 2+5=7$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=2 b=5$$

Tuhia anō te x^{2}+7x+10 hei \left(x^{2}+2x\right)+\left(5x+10\right).

$$\left(x^{2}+2x\right)+\left(5x+10\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(x+2\right)+5\left(x+2\right)$$

Whakatauwehea atu te kīanga pātahi x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+2\right)\left(x+5\right)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$x^{2}+7x+10=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$x=\frac{-7±\sqrt{7^{2}-4\times 10}}{2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 10}}{2}$$

Whakareatia -4 ki te 10.

$$x=\frac{-7±\sqrt{49-40}}{2}$$

Tāpiri 49 ki te -40.

$$x=\frac{-7±\sqrt{9}}{2}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-7±3}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±3}{2} ina he tāpiri te ±. Tāpiri -7 ki te 3.

$$x=-\frac{4}{2}$$

Whakawehe -4 ki te 2.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-7±3}{2} ina he tango te ±. Tango 3 mai i -7.

$$x=-\frac{10}{2}$$

Whakawehe -10 ki te 2.

$$x=-5$$

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te -2 mō te x_{1} me te -5 mō te x_{2}.

$$x^{2}+7x+10=\left(x-\left(-2\right)\right)\left(x-\left(-5\right)\right)$$

Whakamāmātia ngā kīanga katoa o te āhua p-\left(-q\right) ki te p+q.

$$x^{2}+7x+10=\left(x+2\right)\left(x+5\right)$$

Aromātai

$$\left(x+2\right)\left(x+5\right)$$